Universality of actions on $\mathbb HP^2$

TL;DR

This paper proves that any 8-dimensional manifold with a specific circle action shares the same topological invariants as the quaternionic projective plane, implying a form of universality for actions on $ ext{HP}^2$.

Contribution

It establishes that manifolds with a three-fixed-point circle action are topologically and cohomologically equivalent to $ ext{HP}^2$, revealing a universal property of such actions.

Findings

Manifolds with three fixed points have the same weight system as $ ext{HP}^2$

Pontryagin numbers and equivariant cohomology coincide with those of $ ext{HP}^2$

Manifolds with a three-cell decomposition are diffeomorphic to $ ext{HP}^2$

Abstract

We show that any eight-dimensional oriented manifold $M$ possessing smooth circle action with exactly three fixed points has the same weight system as some circle action on $\mathbb HP^2$. It follows that Pontryagin numbers and equivariant cohomology of $M$ coincide to that of $\mathbb HP^2$; if $M$ admits cellular decomposition of only three cells, it is diffeomorphic to $\mathbb HP^2$.